CHAIN CONDITIONS ON NON-SUMMANDS


AYDOĞDU P. , ÖZCAN A. Ç. , SMITH P. F.

JOURNAL OF ALGEBRA AND ITS APPLICATIONS, vol.10, no.3, pp.475-489, 2011 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 10 Issue: 3
  • Publication Date: 2011
  • Doi Number: 10.1142/s0219498811004707
  • Title of Journal : JOURNAL OF ALGEBRA AND ITS APPLICATIONS
  • Page Numbers: pp.475-489

Abstract

Let R be a ring. Modules satisfying ascending or descending chain conditions (respectively, acc and dcc) on non-summand submodules belongs to some particular classes X, such as the class of all R-modules, finitely generated, finite-dimensional and cyclic modules, are considered. It is proved that a module M satisfies acc (respectively, dcc) on non-summands if and only if M is semisimple or Noetherian (respectively, Artinian). Over a right Noetherian ring R, a right R-module M satisfies acc on finitely generated non-summands if and only if M satisfies acc on non-summands; a right R-module M satisfies dcc on finitely generated non-summands if and only if M is locally Artinian. Moreover, if a ring R satisfies dcc on cyclic non-summand right ideals, then R is a semiregular ring such that the Jacobson radical J is left t-nilpotent.